Arabella!
First, let us admire the name: Arabella I.
Please don't make me do the full geneology, which gives me a headache. But Arabella was a descendant of Henry VII through Margaret Tudor, and in 1610 is at least next in line for the thrones of England and Scotland after James I's children Henry, Charles and Elizabeth (the Winter Queen).
She was also considered by Elizabeth to be an option to be heir of England alternative to James, because she was considered to be more "English" than him, even though they are the same degree of relation (they all being descendants of Henry VII). Of course Jimmy's being male gives him the advantage in the succession sweepstakes, however.
Several things about Arabella make her absolutely fascinating for alternate history purposes. One is that no one was quite sure about her religion, although Catholic princes of Europe like Sigismund of Poland were going to great lengths to marry her, making Elizabeth perhaps understandably suspicious.
Perhaps a bit late for the purposes of your departure, Arabella married a member of the Seymour family (yes, as in Jane) who had married a descendant of the Brandon family, who were descended from Henry VII's other daughter, Mary. So conceivably two separate lines with conceivable claims to the throne would be united in their offspring.
Although considering the legendary breeding difficulties of various Tudors and Stuarts, reproduction here is by no means a fait accompli.
As it was, James I imprisoned her for her marriage, which was in the early modern era essentially a national security threat. She died in the Tower after trying to escape England disguised as a man.
She is, in short, completely and thoroughly awesome. Please do have at it if you want to build a timeline around her.